Analytic first-order non-adiabatic coupling matrix elements of spin-adapted open-shell time-dependent density functional theory
While spin-adapted time-dependent density functional theory (TDDFT) approaches significantly improve the excitation energies and gradients of open-shell molecules, the effect of spin-adaptation on non-adiabatic coupling matrix elements (NACMEs) remains unknown for spin-conserving excitations. In this article, we report the derivation, implementation and benchmark studies of the ground state-excited state and excited state-excited state NACMEs of our spin-adapted TDDFT method, X-TDDFT; to our best knowledge, this represents the first implementation of the analytic NACMEs of a spin-adapted TDDFT method. Similar to the X-TDDFT analytic gradients, X-TDDFT NACMEs can be easily implemented on top of an existing U-TDDFT NACME implementation taking into account the restricted open-shell Kohn-Sham (ROKS) reference and the implicit involvement of doubly excited determinants, with acceptable computational overhead. Benchmark calculations reveal that X-TDDFT reduces the error of U-TDDFT NACMEs by 1/3-2/3 (referenced against high-level multireference NACMEs), which leads to large corrections of internal conversion rates (up to two orders of magnitude). In particular, for copper(II) porphyrin, X-TDDFT leads to qualitative revisions of the relative importance of the excited state relaxation pathways, as well as the substituent effects of the internal conversion (IC) rates, suggesting that the error of U-TDDFT NACMEs is not only large but also unsystematic. It is therefore expected that X-TDDFT NACMEs will prove useful in the photophysics/photochemistry studies of open-shell systems such as radicals and transition metal complexes.